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Mirrors > Home > MPE Home > Th. List > oppgsubg | Structured version Visualization version GIF version |
Description: Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
Ref | Expression |
---|---|
oppggic.o | β’ π = (oppgβπΊ) |
Ref | Expression |
---|---|
oppgsubg | β’ (SubGrpβπΊ) = (SubGrpβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 19058 | . . 3 β’ (π₯ β (SubGrpβπΊ) β πΊ β Grp) | |
2 | subgrcl 19058 | . . . 4 β’ (π₯ β (SubGrpβπ) β π β Grp) | |
3 | oppggic.o | . . . . 5 β’ π = (oppgβπΊ) | |
4 | 3 | oppggrpb 19277 | . . . 4 β’ (πΊ β Grp β π β Grp) |
5 | 2, 4 | sylibr 233 | . . 3 β’ (π₯ β (SubGrpβπ) β πΊ β Grp) |
6 | 3 | oppgsubm 19281 | . . . . . . 7 β’ (SubMndβπΊ) = (SubMndβπ) |
7 | 6 | eleq2i 2819 | . . . . . 6 β’ (π₯ β (SubMndβπΊ) β π₯ β (SubMndβπ)) |
8 | 7 | a1i 11 | . . . . 5 β’ (πΊ β Grp β (π₯ β (SubMndβπΊ) β π₯ β (SubMndβπ))) |
9 | eqid 2726 | . . . . . . . . 9 β’ (invgβπΊ) = (invgβπΊ) | |
10 | 3, 9 | oppginv 19278 | . . . . . . . 8 β’ (πΊ β Grp β (invgβπΊ) = (invgβπ)) |
11 | 10 | fveq1d 6887 | . . . . . . 7 β’ (πΊ β Grp β ((invgβπΊ)βπ¦) = ((invgβπ)βπ¦)) |
12 | 11 | eleq1d 2812 | . . . . . 6 β’ (πΊ β Grp β (((invgβπΊ)βπ¦) β π₯ β ((invgβπ)βπ¦) β π₯)) |
13 | 12 | ralbidv 3171 | . . . . 5 β’ (πΊ β Grp β (βπ¦ β π₯ ((invgβπΊ)βπ¦) β π₯ β βπ¦ β π₯ ((invgβπ)βπ¦) β π₯)) |
14 | 8, 13 | anbi12d 630 | . . . 4 β’ (πΊ β Grp β ((π₯ β (SubMndβπΊ) β§ βπ¦ β π₯ ((invgβπΊ)βπ¦) β π₯) β (π₯ β (SubMndβπ) β§ βπ¦ β π₯ ((invgβπ)βπ¦) β π₯))) |
15 | 9 | issubg3 19071 | . . . 4 β’ (πΊ β Grp β (π₯ β (SubGrpβπΊ) β (π₯ β (SubMndβπΊ) β§ βπ¦ β π₯ ((invgβπΊ)βπ¦) β π₯))) |
16 | eqid 2726 | . . . . . 6 β’ (invgβπ) = (invgβπ) | |
17 | 16 | issubg3 19071 | . . . . 5 β’ (π β Grp β (π₯ β (SubGrpβπ) β (π₯ β (SubMndβπ) β§ βπ¦ β π₯ ((invgβπ)βπ¦) β π₯))) |
18 | 4, 17 | sylbi 216 | . . . 4 β’ (πΊ β Grp β (π₯ β (SubGrpβπ) β (π₯ β (SubMndβπ) β§ βπ¦ β π₯ ((invgβπ)βπ¦) β π₯))) |
19 | 14, 15, 18 | 3bitr4d 311 | . . 3 β’ (πΊ β Grp β (π₯ β (SubGrpβπΊ) β π₯ β (SubGrpβπ))) |
20 | 1, 5, 19 | pm5.21nii 378 | . 2 β’ (π₯ β (SubGrpβπΊ) β π₯ β (SubGrpβπ)) |
21 | 20 | eqriv 2723 | 1 β’ (SubGrpβπΊ) = (SubGrpβπ) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 βcfv 6537 SubMndcsubmnd 18712 Grpcgrp 18863 invgcminusg 18864 SubGrpcsubg 19047 oppgcoppg 19261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-subg 19050 df-oppg 19262 |
This theorem is referenced by: lsmmod2 19596 lsmdisj2r 19605 |
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